In the early stage, it's said that radiating loving kindness to opposite sex can potentially arouse lust. While I'm not sure if the person who we radiate loving kindness to can have their lust aroused, is radiating loving kindness in any stage of practice by any means can increase the attachment to that person? What is the proper way to do this?

Practicing loving kindness towards the opposite sex does not increase attachment or arouse lust. But if you are a lustful person, you can easily mistake the counterfeit near enemy as the loving kindness and as a result arouse lust and increase attachment in you.

So if you have an urge to make physical contact with someone or chat up and winover, you should avoid practicing loving kindness towards them until you are skilled enough to identify the counterfeit from authentic Metta.

To develop the sublime abiding of loving-kindness (mettà), you need first
of all be aware that it should not be developed towards a person of the
opposite sex , or a dead person.

A person of the opposite sex should not be used as object, because lust
towards him or her will probably arise. After you have attained jhàna,
however, it is possible to develop loving-kindness towards the opposite
sex as a group with, for example, `May all women be happy.'

Let’s now talk about another category of persons who should not be the object of metta
meditation. When we start practising metta meditation, we should not develop metta for a
person we feel attracted to. The commentary to the Visuddhimagga says that this arouses lust
or tanha. This passage, in fact, refers to metta practice which has not yet become well
established. However, Chanmyay Sayadaw has said that it is possible to practise metta for a
person we have an attraction to without any difficulties when our practice is well developed.
Then it is possible to develop pure and genuine metta.

What is the proper way to do this?

If you are generally spreading Metta as say "May all women be well and happy" then it is fine.

In the early stage, it's said that radiating loving kindness to opposite sex can potentially arouse lust. While I'm not sure if the person who we radiate loving kindness to can have their lust aroused, is radiating loving kindness in any stage of practice by any means can increase the attachment to that person?

This is the reason you should not do it directed at a specific person of the opposite sex. This can lead to lust and attachment.